Interest in the locally compact case is currently very high. Not only because locally compact groups are omnipresent in several areas of mathematics but also because there exists a strict relation between the topological entropy \(h_{top}\)  and the scale function defined in [1]} whenever one considers endomorphisms of totally disconnected locally compact groups (see [2]}, [3]} for details). By analogy with the topological entropy \(h_{top}\)  for totally disconnected locally compact groups, the topological entropy \(\mathrm {ent}^*\)  has been introduced in the category \({}_\mathbb {K}\mathrm {LLC}\)  of locally linearly compact \(\mathbb {K}\) -spaces [4]} whenever \(\mathbb {K}\)  is a discrete field. The main motivation for studying such an entropy function is to reach a better understanding of \(h_{top}\)  by means of the more rigid case of locally linearly compact spaces.
